Algorithmic information distortions and incompressibility in uniform multidimensional networks

TL;DR

This paper investigates the algorithmic information properties of uniform multidimensional networks, revealing distortions and conditions for incompressibility, and compares these properties to classical graph models.

Contribution

It demonstrates the existence of algorithmic information distortions in uniform multidimensional networks and identifies cases of incompressible network families, extending the theoretical understanding of network complexity.

Findings

Existence of algorithmic information distortions in uniform multidimensional networks

Identification of infinite families of incompressible networks

Comparison of multidimensional networks' information content with classical graphs

Abstract

This article presents a theoretical investigation of generalized encoded forms of networks in a uniform multidimensional space. First, we study encoded networks with (finite) arbitrary node dimensions (or aspects), such as time instants or layers. In particular, we study these networks that are formalized in the form of multiaspect graphs. In the context of node-aligned non-uniform (or node-unaligned non-uniform and uniform) multidimensional spaces, previous results has shown that, unlike classical graphs, the algorithmic information of a multidimensional network is not in general dominated by the algorithmic information of the binary sequence that determines the presence or absence of edges. In the present work, first we demonstrate the existence of such algorithmic information distortions for node-aligned uniform multidimensional networks. Secondly, we show that there are particular cases of infinite nesting families of finite uniform multidimensional networks such that each member of these families is incompressible. From these results, we also recover the network topological properties and equivalences in irreducible information content of multidimensional networks in comparison to their isomorphic classical graph counterpart in the previous literature. These results together establish a universal algorithmic approach and set limitations and conditions for irreducible information content analysis in comparing arbitrary networks with a large number of dimensions, such as multilayer networks.